Can I state the Mandelbrot Set/Equation like this?

emergentecon · May 6, 2015

f(z) = z² + c

Raghav Gupta · May 6, 2015

But it must be under iteration. Then definitely you can do that.
Here I suppose you are saying z and c are complex numbers.

emergentecon · May 6, 2015

Raghav Gupta said: ↑

But it must be under iteration. Then definitely you can do that.
Here I suppose you are saying z and c are complex numbers.

Correct yes . . . under iteration, and z + c are complex numbers.
Thanks!

HallsofIvy · May 6, 2015

Just the simple equation "[itex]f(z)= z^2+ c[/itex]" is not a "definition" of anything! The Mandelbrot set is the set of all points, c, in the complex plane such that the recursion "[itex]z_{n+1}= z_n^2+ c[/itex], [itex]z_0= 0[/itex]"

emergentecon · May 6, 2015

HallsofIvy said: ↑

Just the simple equation "[itex]f(z)= z^2+ c[/itex]" is not a "definition" of anything! The Mandelbrot set is the set of all points, c, in the complex plane such that the recursion "[itex]z_{n+1}= z_n^2+ c[/itex], [itex]z_0= 0[/itex]"

I can't say this helps me much.

Is it wrong to say that if you iterate the function f(z) = z^2 + c over the complex numbers, that you get the mandelbrot set?

emergentecon · May 6, 2015

HallsofIvy said: ↑

Just the simple equation "[itex]f(z)= z^2+ c[/itex]" is not a "definition" of anything! The Mandelbrot set is the set of all points, c, in the complex plane such that the recursion "[itex]z_{n+1}= z_n^2+ c[/itex], [itex]z_0= 0[/itex]"

For instance, if I wanted to implement the recursion in Excel, I would in essence specify it as f(z) = z^2 + c

HallsofIvy said: ↑

Just the simple equation "[itex]f(z)= z^2+ c[/itex]" is not a "definition" of anything! The Mandelbrot set is the set of all points, c, in the complex plane such that the recursion "[itex]z_{n+1}= z_n^2+ c[/itex], [itex]z_0= 0[/itex]"

The mandelbrot set is often 'summarised' as z = z^2 + c so why is it wrong to write f(z) = z^2 + c ????

HallsofIvy · May 6, 2015

emergentecon said: ↑

I can't say this helps me much.

Is it wrong to say that if you iterate the function f(z) = z^2 + c over the complex numbers, that you get the mandelbrot set?

??? Just "iterating a function" will give you a lot of numbers and a lot of points where the iteration does not converge to any number. Again, the "Mandlebrot set" is the set of complex numbers for which the iteration does converge.

The mandelbrot set is often 'summarised' as z = z^2 + c so why is it wrong to write f(z) = z^2 + c ????

I don't know what you mean by "summarized" here. What do you do with that iteration to get the Mandlebrot set. (The "Julia sets" use that same iteration but in a different way.)

stedwards · May 6, 2015

Given f(z) = z² + c then z_n+1 = f(z_n). z₀=0, c ranges.

The mandelbrot set is over the domain, c.

If you want to produce a mandelbrot set, you need more than an equations. You need an algorithm.

micromass · May 6, 2015

emergentecon said: ↑

The mandelbrot set is often 'summarised' as z = z^2 + c so why is it wrong to write f(z) = z^2 + c ????

It all depends on context. If you are talking to somebody about dynamical systems, then yes, you can write it like that. If you're not in the correct context, then no, you can't. It all depends on whether the other person will understand you or not.

emergentecon · May 8, 2015

micromass said: ↑

It all depends on context. If you are talking to somebody about dynamical systems, then yes, you can write it like that. If you're not in the correct context, then no, you can't. It all depends on whether the other person will understand you or not.

I mis-stated my question, meant to say his equation, as opposed to the set.
As I know, when he was asked about his work, he wrote down the equation as: z -> z^2 + c
So was wondering if, in this context, z = z^2 + c or f(z) = z^2 + c as opposed to z(n+1) = z(n)^2 + c

Mark44 · May 8, 2015

emergentecon said: ↑

I mis-stated my question, meant to say his equation, as opposed to the set.
As I know, when he was asked about his work, he wrote down the equation as: z -> z^2 + c

I would bet that it was z <- z² + c, with the idea being that, starting with a specified complex number c and some complex number z, you square z, add c, and use that as your new z. Then repeat. And repeat. Ad infinitum.

emergentecon said:

So was wondering if, in this context, z = z^2 + c or f(z) = z^2 + c as opposed to z(n+1) = z(n)^2 + c

The latter formula better captures the idea of an endless sequence of complex numbers.

emergentecon · May 8, 2015

Thanks!

Log in or Sign up

Can I state the Mandelbrot Set/Equation like this?

emergentecon

Raghav Gupta

emergentecon

HallsofIvy

emergentecon

emergentecon

HallsofIvy

stedwards

micromass

emergentecon

Mark44

Staff: Mentor

emergentecon

Mandelbrot Set (Replies: 0)

Mandelbrot Set and Fractals (Replies: 2)

Measure of the Mandelbrot Set (Replies: 6)

Quadratic Recurrence and the Mandelbrot Set (Replies: 1)

Mandelbrot Set and Fractals (Replies: 1)